30 1. Various Ways of Representing Surfaces and Examples
No such uniqueness or classification result is possible with metric
spaces and topological spaces in general; these definitions are exam-
ples of the second category of mathematical objects, and are gener-
alities rather than specifics. In and of themselves, they are far too
general to allow any sort of complete classification or universal un-
derstanding, but they have enough properties to allow us to eliminate
much of the tedious case by case analysis, which would otherwise be
necessary when proving facts about the objects in which we are really
interested. The general notion of a group, or of a Banach space, also
falls into this category of generalities.
Before moving on, there are three definitions of which we ought
to remind ourselves. First, recall that a metric space is complete if
every Cauchy sequence converges. This is not a purely topological
property, since we need a metric in order to define Cauchy sequences;
to illustrate this fact, notice that the open interval (0, 1) and the real
line R are homeomorphic, but that the former is not complete, while
the latter is.
Secondly, we say that a metric space (or subset thereof) is com-
pact if every sequence has a convergent subsequence. In the context of
general topological spaces, this property is known as sequential com-
pactness, and the definition of compactness is given as the require-
ment that every open cover have a finite subcover; for our purposes,
since we will be dealing with metric spaces, the two definitions are
equivalent. There is also a notion of precompactness, which requires
every sequence to have a Cauchy subsequence.
The knowledge that X is compact allows us to draw a number
of conclusions; the most commonly used one is that every continuous
function f : X R is bounded, and in fact achieves its maximum
and minimum. In particular, the product space X × X is compact,
and so the distance function is bounded.
Finally, we say that X is connected if it cannot be written as the
union of non-empty disjoint open sets; that is, if X = A B, with A
and B open and A B = ∅, implies either A = X or B = X. There is
also a notion of path connectedness, which requires for any two points
x, y X the existence of a continuous function f : [0, 1] X such
that f(0) = x and f(1) = y. As is the case with the two forms of
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